Volume 36 • Number 3 • 2013   • Total Colorings of Planar Graphs with Small Maximum Degree Bing Wang, Jian-Liang Wu and Si-Feng Tian Abstract. Let $G$ be a planar graph of maximum degree $\Delta$ and girth $g$, and there is an integer $t(>g)$ such that $G$ has no cycles of length from $g+1$ to $t$. Then the total chromatic number of $G$ is $\Delta+1$ if $(\Delta,g,t)\in\{(5,4,6),(4,4,17)\}$; or $\Delta=3$ and $(g,t)\in\{(5,13),(6,11),(7,11),$ $(8,10),(9,10)\}$, where each vertex is incident with at most one $g$-cycle. 2010 Mathematics Subject Classification: 05C15 Full text: PDF